Best Known (197−37, 197, s)-Nets in Base 4
(197−37, 197, 1539)-Net over F4 — Constructive and digital
Digital (160, 197, 1539)-net over F4, using
- 1 times m-reduction [i] based on digital (160, 198, 1539)-net over F4, using
- trace code for nets [i] based on digital (28, 66, 513)-net over F64, using
- net from sequence [i] based on digital (28, 512)-sequence over F64, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F64 with g(F) = 28 and N(F) ≥ 513, using
- the Hermitian function field over F64 [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F64 with g(F) = 28 and N(F) ≥ 513, using
- net from sequence [i] based on digital (28, 512)-sequence over F64, using
- trace code for nets [i] based on digital (28, 66, 513)-net over F64, using
(197−37, 197, 10878)-Net over F4 — Digital
Digital (160, 197, 10878)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(4197, 10878, F4, 37) (dual of [10878, 10681, 38]-code), using
- discarding factors / shortening the dual code based on linear OA(4197, 16385, F4, 37) (dual of [16385, 16188, 38]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 16385 | 414−1, defining interval I = [0,18], and minimum distance d ≥ |{−18,−17,…,18}|+1 = 38 (BCH-bound) [i]
- discarding factors / shortening the dual code based on linear OA(4197, 16385, F4, 37) (dual of [16385, 16188, 38]-code), using
(197−37, 197, large)-Net in Base 4 — Upper bound on s
There is no (160, 197, large)-net in base 4, because
- 35 times m-reduction [i] would yield (160, 162, large)-net in base 4, but